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Mathematical Function
In mathematics, the Legendre chi function (named after Adrien-Marie Legendre) is a special function whose Taylor series is also a Dirichlet series, given
Legendre_chi_function
Dirichlet beta function Dirichlet L-function Hurwitz zeta function Legendre chi function Lerch transcendent Polylogarithm and related functions: Incomplete
List of mathematical functions
List_of_mathematical_functions
Special function related to the dilogarithm
Dirichlet beta function. The inverse tangent integral is related to the Legendre chi function χ 2 ( x ) = x + x 3 3 2 + x 5 5 2 + ⋯ {\textstyle \chi _{2}(x)=x+{\frac
Inverse_tangent_integral
Special function in mathematics
the Legendre chi function χ ν {\displaystyle \chi _{\nu }} as C ν ( x ) = Re χ ν ( e i x ) {\displaystyle C_{\nu }(x)=\operatorname {Re} \,\chi _{\nu
Hurwitz_zeta_function
Gauss–Legendre algorithm Gauss–Legendre method Gauss–Legendre quadrature Legendre (crater) Legendre chi function Legendre duplication formula Legendre–Papoulis
List of things named after Adrien-Marie Legendre
List_of_things_named_after_Adrien-Marie_Legendre
Operation on formal power series
other series for the zeta-function-related cases of the Legendre chi function, the polygamma function, and the Riemann zeta function include χ 1 ( z ) = ∑
Generating function transformation
Generating_function_transformation
Mathematical approximation of a function
_{n=1}^{\infty }{\frac {1}{n^{3}}}x^{n}\end{aligned}}} The Legendre chi functions are defined as follows: χ 2 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 x
Taylor_series
Special mathematical function
{\operatorname {Ti} _{n}(t)}{t}}dt,} which explains the function name. The Legendre chi function χs(z) (Lewin 1958, Ch. VII § 1.1; Boersma & Dempsey 1992)
Polylogarithm
Probability distribution
distribution with three degrees of freedom). The probability density function (pdf) of the chi-distribution is f ( x ; k ) = { x k − 1 e − x 2 / 2 2 k / 2 −
Chi_distribution
Polynomial sequence
}(x)&=S_{\nu }(1-x).\end{aligned}}} They are related to the Legendre chi function χ ν {\displaystyle \chi _{\nu }} as C ν ( x ) = Re χ ν ( e i x ) S ν ( x )
Bernoulli_polynomials
_{k=1}^{\infty }k^{4}z^{k}={\frac {z(1+z)(1+10z+z^{2})}{(1-z)^{5}}}} The Legendre chi functions are defined as follows: χ 2 ( x ) = ∑ n = 0 ∞ 1 ( 2 n + 1 ) 2 x
List_of_mathematical_series
Partial differential equations
1 2 ( χ ) {\displaystyle Q_{m-{\frac {1}{2}}}(\chi )} is the odd-half-integer degree Legendre function of the second kind, which is a toroidal harmonic
Green's function for the three-variable Laplace equation
Green's_function_for_the_three-variable_Laplace_equation
Special mathematical function
{\tfrac {1}{2}})} The Legendre chi function: χ s ( z ) = ∑ k = 0 ∞ z 2 k + 1 ( 2 k + 1 ) s = z 2 s Φ ( z 2 , s , 1 2 ) {\displaystyle \chi _{s}(z)=\sum _{k=0}^{\infty
Lerch_transcendent
Number-theoretic concept
Stickelberger's theorem. When χ is the Legendre symbol, J ( χ , χ ) = − χ ( − 1 ) = ( − 1 ) p + 1 2 . {\displaystyle J(\chi ,\chi )=-\chi (-1)=(-1)^{\frac {p+1}{2}}\
Jacobi_sum
Gives conditions for the solvability of quadratic equations modulo prime numbers
product of the Riemann zeta function and a certain Dirichlet L-function The Jacobi symbol is a generalization of the Legendre symbol; the main difference
Quadratic_reciprocity
Complex-valued arithmetic function
related branches of mathematics, a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C} } is a Dirichlet character
Dirichlet_character
Sum in algebraic number theory
{\displaystyle G(\chi )=\mu \left({\frac {N}{N_{0}}}\right)\chi _{0}\left({\frac {N}{N_{0}}}\right)G\left(\chi _{0}\right)} where μ is the Möbius function. Consequently
Gauss_sum
Function whose domain is the positive integers
(n)}}.} In this formula ( a p ) {\displaystyle ({\tfrac {a}{p}})} is the Legendre symbol, defined for all integers a and all odd primes p by ( a p ) = {
Arithmetic_function
Set of statistical processes for estimating the relationships among variables
time. The method of least squares was published by Legendre in 1805, and by Gauss in 1809. Legendre and Gauss both applied the method to the problem of
Regression_analysis
Property of functions which is weaker than continuity
convex function. Some operations in convex analysis, such as the Legendre transform automatically produce closed convex functions. The Legendre transform
Semi-continuity
Irreducible representation of the rotation group SO
index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention:
Wigner_D-matrix
Mathematical construct
F(n)=n(n+1)} and χ a Legendre symbol. Here the sum can be evaluated (as −1), a result that is connected to the local zeta-function of a conic section.
Character_sum
Approximation method in statistics
was published by Legendre in 1805. The technique is described as an algebraic procedure for fitting linear equations to data and Legendre demonstrates the
Least_squares
Statistical technique
Practice. Boca Raton: CRC Press. p. 204. ISBN 9781584886167. Legendre, Pierre; Legendre, Louis (2012). Numerical Ecology. Amsterdam: Elsevier. p. 465
Correspondence_analysis
Number, approximately 3.14
continued-fraction representation of the tangent function. French mathematician Adrien-Marie Legendre proved in 1794 that π2 is also irrational. In 1882
Pi
Dutch mathematician
Vooren. Boersma, J.; Dempsey, J.P. (1992). "On the evaluation of Legendre's chi-function" (PDF). Mathematics of Computation. 59 (199): 157–163. doi:10.2307/2152987
Johannes_Boersma
Integer that is a perfect square modulo some integer
extension of the domain is necessary for defining L functions. See Legendre symbol#Properties of the Legendre symbol for examples Lemmermeyer, pp. 111–end Davenport
Quadratic_residue
such that k ( χ ) = k ( γ ) {\displaystyle k(\chi )=k({\sqrt {\gamma }})} . Definition 1. The Legendre symbol ( χ π ) = ε ( π ⊗ χ , 1 / 2 ) ⋅ ε ( π ,
Waldspurger_formula
hypergeometric differential equation. The general solution in terms of Legendre functions of complex index is given by where α = ρ(ρ+1). Further restrictions
Zonal_spherical_function
Part of spectral theory
(1943), "On the representation of an arbitrary function by an integral involving Legendre's functions with a complex index", C. R. Acad. Sci. URSS, 39
Spectral theory of ordinary differential equations
Spectral_theory_of_ordinary_differential_equations
( a ) ζ a {\displaystyle \sum \chi (a)\zeta ^{a}} where χ ( a ) {\displaystyle \chi (a)} here stands for the Legendre symbol (a/p), and the sum is taken
Gaussian_period
Mathematical method in calculus
corresponding to the function of bounded variation χ [ a , b ] ( x ) f ( x ) {\displaystyle \chi _{[a,b]}(x)f(x)} , and functions f ~ , φ ~ {\displaystyle
Integration_by_parts
English mathematician (1907–1969)
zeroes of the local zeta-function immediately imply bounds for sums ∑ χ ( X ( X − 1 ) ( X − 2 ) … ( X − k ) ) {\displaystyle \sum \chi (X(X-1)(X-2)\ldots (X-k))}
Harold_Davenport
Sum type in number theory
\zeta _{p}=\exp(2\pi i/p)} . Equivalently, we can write this using the Legendre symbol as g ( a ; p ) = ∑ n = 0 p − 1 ( 1 + ( n p ) ) ζ p a n . {\displaystyle
Quadratic_Gauss_sum
dynamics and cost functions DNSS point — initial state for certain optimal control problems with multiple optimal solutions Legendre–Clebsch condition
List of numerical analysis topics
List_of_numerical_analysis_topics
Type of fluid flow
)\end{aligned}}} and the P n m {\displaystyle P_{n}^{m}} are the associated Legendre polynomials. The Lamb's solution can be used to describe the motion of
Stokes_flow
Statistical modeling method
the least squares method, which was published by Legendre in 1805, and by Gauss in 1809 ... Legendre and Gauss both applied the method to the problem
Linear_regression
Russian mathematician (1937–2008)
of the form χ ( n ) f ( n ) {\displaystyle \chi (n)f(n)} , where f ( n ) {\displaystyle f(n)} is a function of natural argument. Estimates of that sort
Anatoly_Karatsuba
compatible coefficients. The Pn0 are called Legendre polynomials and the Pnm with m≠0 are called the Associated Legendre polynomials, where subscript n is the
Gravitation_of_the_Moon
Study of collection and analysis of data
analysis. The method of least squares was first described by Adrien-Marie Legendre in 1805, though Carl Friedrich Gauss presumably made use of it a decade
Statistics
Mathematical function of two positive real arguments
transcendental functions (ex, cos x, sin x). Subsequently, many authors went on to study the use of the AGM algorithms. Gauss–Legendre algorithm Generalized
Arithmetic–geometric_mean
Index of articles associated with the same name
theorem to specify which composite numbers are the sums of two squares. Legendre's three-square theorem states which numbers can be expressed as the sum
Sum_of_squares
Function for integral Fourier-like transform
(Also referred to as Daubechies wavelet) Haar wavelet Mathieu wavelet Legendre wavelet Villasenor wavelet Symlet Beta wavelet Hermitian wavelet Meyer
Wavelet
Middle quantile of a data set or probability distribution
the sample median in the early 1800s. However, a decade later, Gauss and Legendre developed the least squares method, which minimizes ( α − α ∗ ) 2 {\displaystyle
Median
numbers greater than 2 {\displaystyle 2} are the sum of two prime numbers. Legendre's conjecture: for every positive integer n {\displaystyle n} , there is
List of unsolved problems in mathematics
List_of_unsolved_problems_in_mathematics
Study of motions and interactions of neutrons
expansion of the angular neutron flux and by assuming that the Legendre polynomials as functions of the neutron direction Ω ^ {\displaystyle {\mathbf {\hat
Neutron_transport
Statistics concept
Gauss–Markov theorem. The least-squares method was published in 1805 by Legendre and in 1809 by Gauss. The first design of an experiment for polynomial
Polynomial_regression
Medical procedure
transplantation. Greenwood Publishing Group. p. 11. ISBN 978-0-313-33542-6. Legendre, Ch; Kreis, H. (November 2010). "A Tribute to Jean Hamburger's Contribution
Kidney_transplantation
Exponentially decreasing bounds on tail distributions of random variables
I=-\log C} . It is equivalent to the Legendre–Fenchel transform or convex conjugate of the cumulant generating function K = log M {\displaystyle K=\log
Chernoff_bound
Geographic coordinate specifying north-south position
axis of a point P on the ellipsoid at latitude ϕ. It was introduced by Legendre and Bessel who solved problems for geodesics on the ellipsoid by transforming
Latitude
Change in energies of a thermodynamic system with respect to particle number
particles are added. A more convenient expression may be obtained by making a Legendre transformation to another thermodynamic potential: the Gibbs free energy
Chemical_potential
Measure of positive and negative charges
Cylindrical multipole moments Spherical multipole moments Laplace expansion Legendre polynomials CP violating moments Many theorists predict elementary particles
Electric_dipole_moment
Design museum in Illinois, United States
Design Pack include Laura Park, Shawna X, Chad Kouri, Susan Kare, Yann Legendre, Paula Scher, Jay Ryan, Mike McQuade, Paul Octavious, Erik Spiekermann
Design_Museum_of_Chicago
doi:10.1038/s42003-026-09824-3. PMC 13125638. PMID 41820599. Benoit, J.; Legendre, L. J.; Araújo, R.; Fernandez, V.; Midzuk, A.; Browning, C.; Abdala, F
2026_in_paleontology
Experimental design that is optimal with respect to some statistical criterion
equivalence theorem in relation to the Legendre-Fenchel conjugacy for convex functions The minimization of convex functions on domains of symmetric positive-semidefinite
Optimal_experimental_design
Quantum
based on the expectation values for some operators, using the theory of Legendre transforms and not semidefinite programming. In some cases, the bounds
Quantum_Fisher_information
2(1-p^{-2})(1-p^{-4})\dots (1-p^{1-n})}} (for n = dim(ƒ) odd) where the Legendre symbol in the second line is interpreted as 0 if p divides 2 det(ƒ). If
Smith–Minkowski–Siegel mass formula
Smith–Minkowski–Siegel_mass_formula
Infectious agent that replicates in cells
(1): 145–55. doi:10.1016/j.virusres.2005.07.011. PMID 16181700. Arslan D, Legendre M, Seltzer V, Abergel C, Claverie JM (October 2011). "Distant Mimivirus
Virus
Earth's most severe extinction event
Retrieved 2024-03-26. Jouault, Corentin; Nel, André; Perrichot, Vincent; Legendre, Frédéric; Condamine, Fabien L. (6 December 2011). "Multiple drivers and
Permian–Triassic extinction event
Permian–Triassic_extinction_event
predicts the orbit of Ceres using a line of best fit 1805 – Adrien-Marie Legendre introduces the method of least squares for fitting a curve to a given set
Timeline of probability and statistics
Timeline_of_probability_and_statistics
Representation theory of the symplectic group
function of compact support equal to 1 near 0, then a ^ = χ a ^ + ( 1 − χ ) a ^ = T + S , {\displaystyle {\widehat {a}}=\chi {\widehat {a}}+(1-\chi ){\widehat
Oscillator_representation
On a distribution yielding the error functions of several well known statistics (1924) presented Pearson's chi-squared test and William Sealy Gosset's
History_of_statistics
German 1777 1855 Invented least squares estimation methods (with Legendre). Used loss functions and maximum-likelihood estimation Quetelet, Adolphe Belgian
Founders_of_statistics
Multivariate statistical technique
PMID 18446182. Dray, S.; Pélissier, R.; Couteron, P.; Fortin, M.-J.; Legendre, P.; Peres-Neto, P. R.; Bellier, E.; Bivand, R.; Blanchet, F. G.; De Cáceres
Spatial Analysis of Principal Components
Spatial_Analysis_of_Principal_Components
Surface Sensitive equivalent
(2l+1)[\exp(2i\delta _{l}(k))-1]P_{l}(\cos \theta ).} Pl(x) is the lth Legendre polynomial, γ is an attenuation coefficient, exp(−2σi2k2) is a Debye–Waller
Surface-extended X-ray absorption fine structure
Surface-extended_X-ray_absorption_fine_structure
Statistical sequence characterizing probability distributions
_{k=0}^{m}(-1)^{m-k}{\binom {m}{k}}{\binom {m+k}{k}}y^{k}} are the shifted Legendre polynomials, orthogonal on [0,1]. In particular λ 1 = ∫ 0 1 Q X ( y ) d
L-moment
Extinct genus of proboscideans
doi:10.1016/j.brainresbull.2006.03.016. PMID 16782503. Benoit, Julien; Legendre, Lucas J.; Tabuce, Rodolphe; Obada, Theodor; Mararescul, Vladislav; Manger
Mastodon
Study of the distribution or space occupied by species
bifurcations and instability. Edge effects Spatial analysis Taylor's law Legendre, P.; Fortin, M.-J. (1989). "Spatial pattern and ecological analysis". Plant
Spatial_ecology
Formulation to quantize gauge field theories in physics
Lagrangian stage, before passing over to Hamiltonian mechanics via the Legendre transformation. The Hamiltonian density is related to the Lie derivative
BRST_quantization
Collins, M.; Mackie, M.; Sakalauskaite, J.; Stiller, J.; Clarke, J. A.; Legendre, L. J.; Douglass, K.; Hansford, J.; Haile, J.; Bunce, M. (2023). "Molecular
2023 in archosaur paleontology
2023_in_archosaur_paleontology
obaf039. doi:10.1093/iob/obaf039. PMC 12690268. PMID 41383558. Byrne, P. J.; Legendre, L. J.; Echols, S.; Farmer, C. G.; Wu, Y.-H.; Huttenlocker, A. K. (2025)
2025 in archosaur paleontology
2025_in_archosaur_paleontology
French naturalized American Christian leader active in France and the United States
C (Oct 1898). "[letter]". Diocese of Fond du Lac. Located at 22bis Rue Legendre in the 17th arrondissement of Paris. Now the Église Saint-Charles-de-Monceau
René_Vilatte
High School, East Setauket, New York JPL · 26948 26950 Legendre 1997 JH10 Adrien-Marie Legendre (1752–1833), French mathematician known for the law of
Meanings of minor-planet names: 26001–27000
Meanings_of_minor-planet_names:_26001–27000
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
Boy/Male
African
God'.
Boy/Male
Arthurian Legend American Hebrew Spanish
Arthur's brother.
Female
Japanese
Variant spelling of Japanese Chou, CHO means "butterfly."
Boy/Male
Indian
Long of time
Boy/Male
Arthurian Legend English Welsh
Arthur's brother.
Boy/Male
Tamil
Aryaman | ஆரà¯à®¯à®®à®¨Â
(Celebrity Name: Amar Upadhyay (Mihir Virani of Kyunki Saas Bhi Kabhi Bahu Thi))
Aryaman | ஆரà¯à®¯à®®à®¨Â
Male
Welsh
 Welsh name, possibly derived from Latin Caius, CAI means "lord." In Arthurian legend, this was the name of a Knight of the Round Table. Compare with another form of Cai.
Boy/Male
Hebrew
Life.
Female
Vietnamese
Vietnamese name CHI means "tree branch."
Surname or Lastname
English
English : variant spelling of Chinn.Chinese : variant of Jin 1.Chinese : Cantonese variant of Qian.Chinese : variant of Qin 1.Chinese : variant of Qin 2.Chinese : variant of Jin 2.Chinese : variant of Jin 3.Korean : there are four Chinese characters for the surname Chin, representing five clans. At least three of the clans have origins in China; most of them migrated to Korea during the Kory{ou} period (ad 918–1392).
Female
Japanese
(æµ) Japanese name CHIE means "wisdom."
Female
Thai/Siamese
Thai name NGAM-CHIT means "good heart."
Boy/Male
Hindu
(Celebrity Name: Amar Upadhyay (Mihir Virani of Kyunki Saas Bhi Kabhi Bahu Thi))
Surname or Lastname
English
English : variant spelling of Gee.Korean : variant of Chi.
Boy/Male
French
Form of Leander. 'Lionlike man.
Male
Scandinavian
 Variant spelling of Scandinavian Kai, possibly CAI means "lord." Compare with another form of Cai.
Boy/Male
French, German, Greek
Lion-man; Form of Leander; Brave as a Lion
Boy/Male
Muslim
Long of time
Boy/Male
Sikh
Source of light
Female
Vietnamese
Vietnamese name THI means "poem."
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
Boy/Male
Arabic, Australian
Name
Boy/Male
Danish, German, Swedish
Strong; Noble
Boy/Male
Swedish American English German
Bear.
Female
Hindi/Indian
(विमला) Feminine form of Hindi Vimal, VIMALA means "clean, pure."
Girl/Female
Indian, Telugu
Good Gem
Boy/Male
Tamil
Lord Rama
Boy/Male
Indian
Good
Boy/Male
Sikh
Friend of the Lord Sun
Boy/Male
Muslim/Islamic
Commander Prince, Khalifah
Girl/Female
Gujarati, Hindu, Indian
The Consort of God Krishna
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
LEGENDRE CHI-FUNCTION
n.
A univalent hydrocarbon radical of the ethylene series, CH2:CH; -- called also vinyl. See Vinyl.
n.
A chip; a alice.
n.
One whi gives evidence.
n.
A story or legend abounding in miracles.
n.
One whi sips.
n.
Formerly, the radical methyl, CH3.
n.
A child or babe; as, a forward chit; also, a young, small, or insignificant person or animal.
v. t.
To tell or narrate, as a legend.
v. i.
Alt. of Degener
p. pr. & vb. n.
of Chip
a.
Pertaining to deeds or feats of arms; legendary.
n.
One who relates legends.
n.
The unsymmetrical hypothetical hydrocarbon radical, CH3.CH2.CH, analogous to ethylidene, and regarded as the type of certain derivatives of propane; -- called also propylidene.
a.
Having a smooth chin; beardless.
n.
A book of legends; a tale or narrative.
imp. & p. p.
of Chip
a.
Former; previous; of times gone by; as, a ci-devant governor.
a.
Of or pertaining to a legend or to legends; consisting of legends; like a legend; fabulous.
n.
A reciter of gests or legendary tales; a story-teller.